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Mirrors > Home > MPE Home > Th. List > Mathboxes > imaiun1 | Structured version Visualization version GIF version |
Description: The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.) |
Ref | Expression |
---|---|
imaiun1 | ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom4 3212 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) | |
2 | vex 3444 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 2 | elima3 5903 | . . . . 5 ⊢ (𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) |
4 | 3 | rexbii 3210 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) |
5 | eliun 4885 | . . . . . . 7 ⊢ (〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵) | |
6 | 5 | anbi2i 625 | . . . . . 6 ⊢ ((𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ (𝑧 ∈ 𝐶 ∧ ∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵)) |
7 | r19.42v 3303 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵) ↔ (𝑧 ∈ 𝐶 ∧ ∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵)) | |
8 | 6, 7 | bitr4i 281 | . . . . 5 ⊢ ((𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) |
9 | 8 | exbii 1849 | . . . 4 ⊢ (∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) |
10 | 1, 4, 9 | 3bitr4ri 307 | . . 3 ⊢ (∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶)) |
11 | 2 | elima3 5903 | . . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)) |
12 | eliun 4885 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶)) | |
13 | 10, 11, 12 | 3bitr4i 306 | . 2 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶)) |
14 | 13 | eqriv 2795 | 1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∃wrex 3107 〈cop 4531 ∪ ciun 4881 “ cima 5522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-iun 4883 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 |
This theorem is referenced by: trclimalb2 40427 |
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